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31: 35.4 Partitions and Zonal Polynomials

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35.4.9

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

32: 14.3 Definitions and Hypergeometric Representations

… ►In terms of the Gegenbauer function $C_{\alpha }^{(\beta )}\left(x\right)$ and the Jacobi function ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$ (§§15.9(iii), 15.9(ii)): … ►

14.3.23

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$: Jacobi function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv)

Permalink:

http://dlmf.nist.gov/14.3.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

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33: 18.15 Asymptotic Approximations

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18.15.1 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-1}{2}^{2n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1)\left(\sum _{m=0}^{M-1}\frac{f_m(\theta )}{2^m{\left(2n+\alpha +\beta +2\right)}_m}+O\left(n^{-M}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $f_m(\theta )$

Referenced by:

§18.15(i), §18.15(i)

Permalink:

http://dlmf.nist.gov/18.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►Also, $\mathrm{B}(a,b)$ is the beta function (§5.12) and ►

18.15.2 $$f_m(\theta )=\sum _{\mathrm{\ell }=0}^m\frac{C_{m,\mathrm{\ell }}(\alpha ,\beta )}{\mathrm{\ell }!(m-\mathrm{\ell })!}\frac{\cos {\theta }_{n,m,\mathrm{\ell }}}{{\left(\sin \frac{1}{2}\theta \right)}^{\mathrm{\ell }}{\left(\cos \frac{1}{2}\theta \right)}^{m-\mathrm{\ell }}},$$

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Defines:

$f_m(\theta )$ (locally)

Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $C_{m,\mathrm{\ell }}(\alpha ,\beta )$ and ${\theta }_{n,m,\mathrm{\ell }}$

Permalink:

http://dlmf.nist.gov/18.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.4_5 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-\frac{1}{2}}{n}^{-\frac{1}{2}}\cos \left(\frac{1}{2}(2n+\alpha +\beta +1)\theta -\frac{1}{4}(2\alpha +1)\pi \right)+O\left(n^{-\frac{3}{2}}\right),$$ $\alpha ,\beta \in ℝ$,

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\in$: element of, $ℝ$: real line, $\sin z$: sine function and $n$: nonnegative integer

Proveds:

Ismail (2009, Theorem 4.3.1); Szegő (1975, Theorem 8.21.8); with attribution to Darboux(proved)

Referenced by:

§18.15(i), Erratum (V1.2.0) Section 18.15

Permalink:

http://dlmf.nist.gov/18.15.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.15(i), §18.15 and Ch.18

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18.15.6 $${(\sin \frac{1}{2}\theta )}^{\alpha +\frac{1}{2}}{(\cos \frac{1}{2}\theta )}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)=\frac{\mathrm{\Gamma }\left(n+\alpha +1\right)}{2^{\frac{1}{2}}{\rho }^{\alpha }n!}\left({\theta }^{\frac{1}{2}}{J}_{\alpha }\left(\rho \theta \right)\sum _{m=0}^M\frac{A_m(\theta )}{{\rho }^{2m}}+{\theta }^{\frac{3}{2}}{J}_{\alpha +1}\left(\rho \theta \right)\sum _{m=0}^{M-1}\frac{B_m(\theta )}{{\rho }^{2m+1}}+{\epsilon }_M(\rho ,\theta )\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $\rho$ , ${\epsilon }_M(\rho ,\theta )$ , $A_m(\theta )$: coefficient and $B_m(\theta )$: coefficient

Permalink:

http://dlmf.nist.gov/18.15.E6

Encodings:

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See also:

Annotations for §18.15(i), §18.15 and Ch.18

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34: Bibliography G

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Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.

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Notes:

Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.

External Links:

ISSN 0001-0782, Document, Certification

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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35: Bibliography K

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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.

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External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

… ►

Orthogonal polynomials with weight function ${(1-x)}^{\alpha }{(1+x)}^{\beta }+M\delta (x+1)+N\delta (x-1)$ . Canad. Math. Bull. 27 (2), pp. 205–214.

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External Links:

ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt

Referenced by:

§18.36(i), §18.36(i).

Encodings:

BibTeX

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36: 24.9 Inequalities

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24.9.9 $$\beta =2+\frac{\ln \left(1-6{\pi }^{-2}\right)}{\ln 2}=0.6491\mathrm{\dots }.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $\beta$: quantity

Permalink:

http://dlmf.nist.gov/24.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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37: 18.17 Integrals

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18.17.9 $$\frac{{(1-x)}^{\alpha +\mu }{P}_n^{(\alpha +\mu ,\beta -\mu )}\left(x\right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}={\int }_x^1\frac{{(1-y)}^{\alpha }{P}_n^{(\alpha ,\beta )}\left(y\right)}{\mathrm{\Gamma }\left(\alpha +n+1\right)}\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§1.15(vi), §18.17(iv), §18.17(iv), §18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.10 $$\frac{x^{\beta +\mu }{(x+1)}^n}{\mathrm{\Gamma }\left(\beta +\mu +n+1\right)}{P}_n^{(\alpha ,\beta +\mu )}\left(\frac{x-1}{x+1}\right)={\int }_0^x\frac{y^{\beta }{(y+1)}^n}{\mathrm{\Gamma }\left(\beta +n+1\right)}{P}_n^{(\alpha ,\beta )}\left(\frac{y-1}{y+1}\right)\frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, $x>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.11 $$\frac{\mathrm{\Gamma }\left(n+\alpha +\beta -\mu +1\right)}{x^{n+\alpha +\beta -\mu +1}}{P}_n^{(\alpha ,\beta -\mu )}\left(1-2x^{-1}\right)={\int }_x^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{y^{n+\alpha +\beta +1}}{P}_n^{(\alpha ,\beta )}\left(1-2y^{-1}\right)\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\alpha +\beta +1>\mu >0$, $x>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§1.15(vi), §18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.16 $${\int }_{-1}^1{(1-x)}^{\alpha }{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{{(\mathrm{i}y)}^n{\mathrm{e}}^{\mathrm{i}y}}{n!}{2}^{n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1){}_1{}^{}F_1^{}(n+\alpha +1;2n+\alpha +\beta +2;-2\mathrm{i}y).$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), §18.17(vi)

Permalink:

http://dlmf.nist.gov/18.17.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. …

38: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.5 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(c-a\right)>\frac{1}{2}(m-1)$, .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $c$: complex variable, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(iv)

Permalink:

http://dlmf.nist.gov/35.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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39: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.13 $\mathrm{\Re }\left(b_1-a_1\right),\mathrm{\Re }\left(a_1\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $p$: nonnegative integer, $q$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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40: Bibliography I

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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